Basically the mechanisms of muscle growth are what sickstan described, with these clarifications:
1) There's sarcomere ("functional") hypertrophy (growth of the contractile components of the muscle) and sarcoplasmic ("non-functional") hypertrophy (growth of the sarcoplasm and the sarcoplasmic reticulum). The second is lost more easily than the first.
2) For all practical purposes you can forget hyperplasia. Yes, it does happen, there are studies, yada, yada, yada... but it's negligible - at least for the very vast majority of people. So no increase in number of muscle fibers.
Now, there is a passive contraction of the muscles which we refer to as "tone". In two persons with the same muscle mass and the same bodyfat levels one of which exercises and the other doesn't, we are *maybe* going to feel the first person's muscles a little "harder" because exercise *may* increase that passive contraction. But this doesn't make much of a difference anyway.
There is also the age-related muscle loss that is called
sarcopenia but we are not interested in that.
The normal muscle atrophy that lack of exercise brings has to do mainly with normalization of anabolic hormones. Exercise increases these hormones above normal levels. When exercise stops these hormones return to normal. E.g., your normal testosterone levels are not able to support as much muscle as your exercise-induced testosterone levels. The exact mechanisms of how various hormone levels affect muscle turnover are rather well documented. There are other atrophy mechanisms as well (metabolic, mechanical, neurological) but the hormonal ones are the most important. Those hormonal differences are responsible anyway for the vastly different muscle masses of people who have never exercised.
However, this atrophy does not affect significantly the "muscle tone" and the flabbiness of muscles that most people perceive as the effects of exercise cessation (barring the aforementioned residual muscle tone). What happens is that in most cases exercise cessation results in an increase of body fat percentage because of less energy expenditure and relaxed dietary habits. If you feel the muscles of people who are sedentary yet have very low bodyfat levels, you'll see they are rather hard despite the lack of exercise (keep in mind here that "low bodyfat levels" isn't necessarily the same as "thin" since many "thin" people don't have particularly low bf levels). If someone who stops exercise were to re-adjust his caloric intake to suit his new needs and maintain adequate protein in his diet, then his muscles would atrophy but would probably not feel any "flabbier". We also have to differentiate according to the type of exercise since a long-distance runner would probably not lose any muscle after stopping running.
That's for "muscle tone", strength is a rather different story.